System and method to solve dynamic multi-factor models in finance

ABSTRACT

Methods and systems for estimating time-varying factor exposures of either an individual financial instrument or a portfolio of such instruments, through the solution of a constrained multi-criteria dynamic optimization problem, providing an estimation error function and one or more transition error functions to be minimized over a period of time. The factor exposures relay the influence of the factors on the return of the instrument or portfolio. The estimation error function provides the estimation error at each time interval between the return of the asset collection and a sum of products of each factor exposure and its respective factor. Each transition error function provides a transition error of each factor exposure between time intervals. In one embodiment, the constraints can include a budget constraint and non-negativity bounds applying to some or all of the factor exposures. In other embodiments, the method and system can be applied to estimating any time-varying weight that is used in a model, to relay the influence of one or more independent variables on a dependent financial or economic variable, through the solution of a constrained multi-criteria dynamic problem, minimizing estimation error and transition error terms. In other embodiments, the solution of a multi-criteria dynamic problem can be used as part of a method and system to determine structural breakpoints for each factor, and also as part of a method and system for determining optimal parameters to weight the transition error functions and selecting the factors included in the model.

PRIORITY CLAIM

The present application is a Continuation application of U.S. patentapplication Ser. No. 10/431,838 filed on May 7, 2003 entitled “Methodand System to Solve Dynamic Multi-Factor Models in Finance” now U.S.Pat. No. 7,617,142 issued on Nov. 10, 2009 which claims priority to U.S.Provisional Patent Application Ser. No. 60/378,562 filed on May 7, 2002,the entire disclosure of both applications are expressly incorporatedherein by reference.

FIELD OF THE INVENTION

The present invention relates generally to systems and methods forestimating time-varying factor exposures in financial or economic modelor problem, through the solution of a multi-factor dynamic optimizationof the model or problem, while meeting the constraints for the estimatedtime-varying factor exposures in the model or problem.

BACKGROUND OF THE INVENTION

The following references, discussed and/or cited in this application,are hereby expressly incorporated herein by reference in their entiretyinto this application;

-   1. Sharpe, William F., Capital asset prices: A theory of market    equilibrium under conditions of risk. Journal of Finance, September    1964;-   2. Chen, Nai-fu, Roll, Richard, Ross, Stephen A., Economic forces    and the stock market. Journal of Business, 59, Jul. 1986;-   3. Rosenberg, B., Choosing a multiple factor model. Investment    Management Review, November/December 1987;-   4. Sharpe, William F., Determining a Fund's Effective Asset Mix.    Investment Management Review, November/December 1988;-   5. Sharpe, William F., Asset Allocation: Management Style and    Performance Measurement. The Journal of Portfolio Management, Winter    1992;-   6. Kalaba, R., Tesfatsion, L., Time-Varying Linear Regression via    Flexible Least Squares. Computers and Mathematics with Applications,    17, 1989;-   7. Kalaba, R., Tesfatsion, L., Flexible least squares for    approximately linear systems. IEEE Transactions on Systems, Man, and    Cybernetics, SMC-5, 1990;-   8. Tesfatsion, L., GFLS implementation in FORTRAN and the algorithm.    http://www.econ.iastate.edu/tesfatsi/gflshelp.htm (1997);-   9. Lütkepohl, H., Herwartz, H., Specification of varying coefficient    time series models via generalized flexible least squares. Journal    of Econometrics, 70, 1996;-   10. Wright, S., Primal-dual interior-point methods, SIAM, 1997; and-   11. Stone, M., Cross-validatory choice and assessment of statistical    predictions. Journal of Royal Statistical Soc., B 36, 1974.

A. Multi-Factor Models in Finance

Factor models are well known in finance, among them a multi-indexCapital Asset Prices Model (CAPM) and Arbitrage Pricing Theory (APT).These models allow for a large number of factors that can influencesecurities returns.

The multi-factor CAPM, for example, described in Sharpe, William F.,Capital asset prices: A theory of market equilibrium under conditions ofrisk. Journal of Finance, September 1964, pp. 425-442, can berepresented by the equation:r−r ^((f))≅α+β⁽¹⁾(r ⁽¹⁾ −r ^((f)))+β⁽²⁾(r ⁽²⁾ −r ^((f)))+ . . .+β^((n))(r ^((n)) −r ^((f)))  (1)where r is the investment return (security or portfolio of securities),r^((i)) are returns on the market portfolio as well as changes in otherfactors like inflation, and r^((f)) is return on a risk-free instrument.

In the multi-factor APT model (described, for example, in Chen, N.,Richard R., Stephen A. R., Economic forces and the stock market. Journalof Business, 59, July 1986, pp. 383-403):r≅{dot over (α)}+β ⁽¹⁾ I ⁽¹⁾+β⁽²⁾ I ⁽²⁾+ . . . +β^((n)) I ^((n))  (2)the factors I^((i)) are typically chosen to be the major economicfactors that influence security returns, like industrial production,inflation, interest rates, business cycle, etc. (described, for example,in Chen, N., Richard R., Stephen A. R., Economic forces and the stockmarket. Journal of Business, 59, July 1986, pp. 383-403, and inRosenberg, B., Choosing a multiple factor model. Investment ManagementReview, November/December 1987, pp. 28-35).

Coefficients β⁽¹⁾, . . . , β^((n)) in the CAPM (1) and APT (2) modelsare called factor exposures. Along with the constant α, the factorexposures make the vector of model parameters (α,β⁽¹⁾, . . . , β^((n))),which is typically estimated by applying a linear regression techniqueto the time series of security/portfolio returns r_(t) and economicfactors r_(t) ^((i)) or I_(t) ^((i)) over a certain estimation windowt=1, . . . , N:

$\begin{matrix}{\left( {\hat{\alpha},{\hat{\beta}}^{(1)},\ldots\mspace{14mu},{\hat{\beta}}^{(n)}} \right) = {\underset{\alpha,\beta^{(1)},\ldots,\beta^{(n)}}{\arg\mspace{20mu}\min}{\sum\limits_{t = 1}^{N}{\left( {r_{t} - \alpha - {\beta^{(1)}I_{t}^{(1)}} - \ldots - {\beta^{(n)}I_{t}^{(n)}}} \right)^{2}.}}}} & (3)\end{matrix}$

One of the most effective multi-factor models for analyses of investmentportfolios, called the Returns Based Style Analysis (RBSA), wassuggested by Prof. William F. Sharpe (for example, in Sharpe, WilliamF., Determining a Fund's Effective Asset Mix. Investment ManagementReview, November/December 1988, pp. 59-69, and in Sharpe, William F.,Asset Allocation: Management Style and Performance Measurement. TheJournal of Portfolio Management, Winter 1992, pp. 7-19). In the RBSAmodel, the periodic return y of a portfolio consisting of n kinds ofassets is approximately represented by a linear combination of singlefactors (x⁽¹⁾, . . . , x^((n))) whose role is played by periodic returnsof generic market indices for the respective classes of assets. Toenhance the quality of parameter estimation, a set of linear constraintsis added to the basic equation:

$\begin{matrix}{{y \cong {\alpha + {\beta^{(1)}x^{(1)}} + {\beta^{(2)}x^{(2)}} + \ldots + {\beta^{(n)}x^{(n)}}}},{{\sum\limits_{i = 1}^{n}\beta^{(i)}} = 1},{\beta^{(i)} \geq 0},{i = 1},\ldots\mspace{11mu},{n.}} & (4)\end{matrix}$

In such a model, x^((i)), i=1, . . . , n, represent periodic returns(for example, daily, weekly or monthly) of generic market indices suchas bonds, equities, economic sectors, country indices, currencies, etc.For example (as described in Sharpe, William F., Asset Allocation:Management Style and Performance Measurement. The Journal of PortfolioManagement, Winter 1992, pp. 7-19), twelve such generic asset indicesare used to represent possible areas of investment.

To estimate the parameters of equation (4), Sharpe used the ConstrainedLeast Squares Technique, i.e., the parameters are found by solving theconstrained quadratic optimization problem in a window of t=1, . . . , Ntime periods in contrast to the unconstrained one (3):

$\begin{matrix}\left\{ \begin{matrix}{{\left( {\hat{\alpha},{\hat{\beta}}^{(1)},\ldots\mspace{14mu},{\hat{\beta}}^{(n)}} \right) = {\underset{\alpha,\beta^{(1)},\ldots,\beta^{(n)}}{\arg\mspace{14mu}\min}{\sum\limits_{t = 1}^{N}\left( {y_{t} - \alpha - {\beta^{(1)}x_{t}^{(1)}} - \ldots - {\beta^{(n)}x_{t}^{(n)}}} \right)^{2}}}},} \\{{{{subject}\mspace{14mu}{to}\mspace{14mu}{\sum\limits_{i = 1}^{n}\beta^{(i)}}} = 1},{\beta^{(i)} \geq 0},{i = 1},\ldots\mspace{14mu},{n.}}\end{matrix} \right. & (5)\end{matrix}$

Model parameters (α,β⁽¹⁾, . . . , β^((n))) estimated using unconstrained(3) and constrained least squares techniques (5) represent averagefactor exposures in the estimation window—time interval t=1, . . . , N.However, the factor exposures typically change in time. For example, anactive trading of a portfolio of securities can lead to significantchanges in its exposures to market indices within the interval.Detecting such dynamic changes, even though they happened in the past,represents a very important task.

In order to estimate dynamic changes in factor exposures, a movingwindow technique is typically applied. For example, in RBSA model (4),the exposures at any moment of time t are determined on the basis ofsolving (5) using a window of K portfolio returns [t−(K−1), . . . , t]and the returns on asset class indices over the same time period (asdescribed, for example, in 5. Sharpe, William F., Asset Allocation:Management Style and Performance Measurement. The Journal of PortfolioManagement, Winter 1992, pp. 7-19):

$\begin{matrix}\left\{ \begin{matrix}{{\left( {{\hat{\alpha}}_{t},{\hat{\beta}}_{t}^{(1)},\ldots\mspace{14mu},{\hat{\beta}}_{t}^{(n)}} \right) = {\underset{\alpha,\beta^{(1)},\ldots,\beta^{(n)}}{\arg\mspace{14mu}\min}{\sum\limits_{\tau = 0}^{K - 1}\left( {y_{t - \tau} - \alpha - {\beta^{(1)}x_{t - \tau}^{(1)}} - \ldots - {\beta^{(n)}x_{x_{t - \tau}}^{(n)}}} \right)^{2}}}},} \\{{{{subject}\mspace{14mu}{to}\mspace{14mu}{\sum\limits_{i = 1}^{n}\beta^{(i)}}} = 1},{\beta^{(i)} \geq 0},{i = 1},\ldots\mspace{14mu},n,}\end{matrix} \right. & (6)\end{matrix}$

By moving such estimation window forward period by period, dynamicchanges in factor exposures can be approximately estimated.

The moving window technique described above has its limitations anddeficiencies. The problem setup assumes that exposures are constantwithin the window, yet it is used to estimate their changes. Reliableestimates of model parameters can be obtained only if the window issufficiently large which makes it impossible to sense changes thatoccurred within a day or a month, and, therefore, such technique can beapplied only in cases where parameters do not show marked changes withinit: (α_(s), β_(s) ⁽¹⁾, . . . , β_(s) ^((n)))≅const, t−(K−1)≦s≦t. Inaddition, such approach fails to identify very quick, abrupt changes ininvestment portfolio exposures that can occur due to trading.

In situations, where detecting dynamic exposures represents an importanttask, the widow technique is inadequate, and a fundamentally newapproach to estimating multi-factor models with changing properties arerequired. It is just the intent of this patent to fill in this gap.

B. The Dynamic RBSA Model

The multi-factor RBSA model (4), as well as the CAPM (1) and APT ones(2), are, in their essence, linear regression models with constantregression coefficients (α,β⁽¹⁾, . . . , β^((n))).

In order to monitor a portfolio for quick changes in investmentallocation or investment style, deviations from investment mandate,etc., a dynamic regression RBSA model is needed to represent the timeseries of portfolio returns y_(t) as dynamically changing linearcombination of a finite number n of time series of basic factorsx_(t)=(x_(t) ⁽¹⁾, . . . , x_(t) ^((n)))^(T) with unknown real-valuedfactor exposures β_(t)=(β_(t) ⁽¹⁾, . . . , β_(t) ^((n)))^(T) and anunknown auxiliary term α_(t). However, in the RBSA model, both thefactor exposures and the intercepts are subject to appropriateconstraints (α_(t),β_(t))εZ, in the simplest case, the linear ones

${{\sum\limits_{i = 1}^{n}\beta_{t\;}^{(i)}} = 1},{\beta_{t}^{(i)} \geq 0.}$

$\begin{matrix}\left\{ \begin{matrix}{{y_{t} = {{\alpha_{t} + {\sum\limits_{i = 1}^{n}{\beta_{t}^{(i)}x_{t}^{(i)}}} + e_{t}} = {\alpha_{t} + {\beta_{t}^{T}x_{t}} + e_{t}}}},} \\{{{\left( {\alpha_{t},\beta_{t}} \right) \in Z},}\mspace{310mu}}\end{matrix} \right. & (7)\end{matrix}$where e_(t) is the residual model inaccuracy treated as white noise.

Note that unlike (5) and (6), the model (7) assumes that factorexposures are changing in every period or time interval t. The presentinvention specifies constraints (α_(t),β_(t))εZ adequate to most typicalproblems of financial management, and describes a general way ofestimating dynamic multi-factor models under those constraints.

C. Insufficiency of Existing Methods of Estimating Dynamic Linear Models

i. Flexible Least Squares (FLS)

A method of unconstrained parameter estimation in dynamic linearregression models was suggested by Kalaba and Tesfatsion under the nameof Flexible Least Squares (FLS) method, as described, for example, inKalaba, R., Tesfatsion, L., Time-Varying Linear Regression via FlexibleLeast Squares. Computers and Mathematics with Applications, 17, 1989,pp. 1215-1245, in Kalaba, R., Tesfatsion, L., Flexible least squares forapproximately linear systems. IEEE Transactions on Systems, Man, andCybernetics, SMC-5, 1990, 978-989, and in Tesfatsion, L., GFLSimplementation in FORTRAN and the algorithm, athttp://www.econ.iastate.edu/tesfatsi/gflshelp.htm (1997). To estimatethe succession of unknown n-dimensional regression coefficient vectors(β_(t), t=1, . . . , N) under the assumption that (y_(t), t=1, . . . ,N) and (x_(t), t=1, . . . , N) are known time series, it was proposed tominimize the quadratic objective function

$\begin{matrix}{{\left( {{\hat{\beta}}_{t},{t = 1},\ldots\mspace{14mu},N} \right) = {\underset{\beta_{t},{t = 1},\mspace{14mu}\ldots\mspace{14mu},N}{\arg\mspace{14mu}\min}\left\lbrack {{\sum\limits_{t = 1}^{N}\left( {y_{t} - {x_{t\;}^{T}\beta_{t}}} \right)^{2}} + {\lambda{\sum\limits_{t = 2}^{N}{\left( {\beta_{t} - {V\;\beta_{t - 1}}} \right)^{T}{U\left( {\beta_{t} - {V\;\beta_{t - 1}}} \right)}}}}} \right\rbrack}},} & (8)\end{matrix}$where V and U are known (n×n) matrices, where matrix V expresses theassumed linear transition model of the hidden dynamics of time-varyingregression coefficients, and matrix λU, λ>0, is responsible for thedesired smoothness of the sought-for succession of estimates({circumflex over (β)}_(t), t=1, . . . , N). In practice, the transitionmatrix V is considered to be the identity matrix.

The structure of the criterion (8) explicitly displays the essence ofthe FLS approach to the problem of parameter estimation in dynamiclinear regression models as a multi-objective optimization problem. Thefirst term is the squared Euclidean norm of the linear regressionresiduals ∥e_([1, . . . , N])∥, e_(t)=y_(t)−x_(t) ^(T)β_(t), responsiblefor the model fit, the second term is a specific squared Euclidean normof the variation of model parameters ∥w_([2, . . . , N)]∥,w_(t)=(β_(t)−Vβ_(t−1))^(T)U(β_(t)−Vβ_(t−1)), which is determined by thechoice of the positive semidefinite matrix U, whereas the positiveweighting coefficient λ is to be chosen to balance the relative weightsbetween these two particular objective functions. If λ→∞, the solutionof (8) becomes very smooth and approaches the ordinary least squaressolution, while selecting λ close to zero makes the parameters veryvolatile. Typically, the equation (8) is solved and presented fordifferent values of parameter λ.

ii. Generalized Flexible Least Squares (GFLS)

A generalization of the FLS method was suggested by Lütkepohl andHerwartz under the name of Generalized Flexible Least Squares (GFLS)method, for example, in Lütkepohl, H., Herwartz, H., Specification ofvarying coefficient time series models via generalized flexible leastsquares. Journal of Econometrics, 70, 1996, pp. 261-290), and presentedas follows:

$\begin{matrix}\left\{ \begin{matrix}{\left( {{\hat{\beta}}_{t},{t = 1},\ldots\mspace{14mu},N} \right) =} \\{{\underset{\beta_{t},{t = 1},\ldots,N}{\arg\mspace{14mu}\min}\left\lbrack {{\sum\limits_{t = 1}^{N}\left( {y_{t} - {x_{t}^{T}\beta_{t}}} \right)^{2}} + {\lambda_{1}{\sum\limits_{t = {k + 1}}^{N}{\left( {\beta_{t} - {\overset{¨}{\beta}}_{1,t}} \right)^{T}{U_{1}\left( {\beta_{t} - {\overset{¨}{\beta}}_{1,t}} \right)}}}} + {\lambda_{2}{\sum\limits_{t = {s + 1}}^{N}{\left( {\beta_{t} - {\overset{¨}{\beta}}_{2,t}} \right)^{T}{U_{2}\left( {\beta_{t} - {\overset{¨}{\beta}}_{2,t}} \right)}}}}} \right\rbrack},} \\{{{\overset{¨}{\beta}}_{1,t} = {{V_{1,1}\beta_{t - 1}} + \ldots + {V_{1,k}\beta_{t - k}}}},} \\{{\overset{¨}{\beta}}_{2,t} = {V_{2}{\beta_{t - s}.}}}\end{matrix} \right. & (9)\end{matrix}$

In this specific version of the multi-objective criterion, two differentnorms of the model parameter variation are fused, namely, the norm basedon a higher-order model of parameter dynamics ∥w_(1,[k+1, . . . , N])∥,w_(1,t)=(β_(t)−{umlaut over (β)}_(1,t))^(T)U₁(β_(t)−{umlaut over(β)}_(1,t)), and that representing the variation at a single predefinedvalue of the time lag ∥w_(2,[s+1, . . . , N])∥, w_(2,t)=(β_(t)−{umlautover (β)}_(2,t))^(T)U₂(β_(t)−{umlaut over (β)}_(2,t)). Each of thesenorms is defined by the choice of the respective positive semidefinitematrix, respectively, U₁ and U₂.

Algorithms for solving the FLS (8) and GFLS (9) problems were described,for example, in Tesfatsion, L., GFLS implementation in FORTRAN and thealgorithm. http://www.econ.iastate.edu/tesfatsi/gflshelp.htm (1997), andin Lütkepohl, H., Herwartz, H., Specification of varying coefficienttime series models via generalized flexible least squares. Journal ofEconometrics, 70, 1996, pp. 261-290.

However, the FLS and GFLS methods discussed above, never mention,suggest or otherwise describe methods or systems for estimating dynamicmulti-factor models adequate for financial applications, first of all,because of the presence of constraints (α₁,β₁)εZ in the RBSA model (7)and other financial or economic models or problems. The methods also donot mention, suggest or otherwise describe methods or systems fordetermining structural breakpoints with a multi-factor dynamicoptimization problem or determining cross validation statistics toformulate the model or problem. The present invention provides methodsand systems for resolving these and other issues arising in financial oreconomic applications.

SUMMARY OF INVENTION

The present invention provides methods and systems for estimatingtime-varying factor exposures in models or problems, such as, forexample, in the RBSA model and other financial and economic models orproblems, through a multi-factor dynamic optimization of the models orproblems, while meeting the constraints for the estimated time-varyingfactor exposures.

One embodiment of the present invention describes a method of estimatingtime-varying weights (for example, factor exposures) for independentvariables (such as, e.g., factors or indexes) at each time interval in aperiod of time, through a dynamic optimization of a model relating aninfluence of one or more independent variables on a dependent economicvariable. The method includes the steps of: receiving data related tothe dependent economic variable for all the time intervals, receivingdata related to the independent variables for all the time intervals,and, determining at least one weight for one independent variable ateach time interval, that minimizes values of two or more objectivefunctions while meeting at least one constraint on possible values forthe weight. The constraints and objective functions are formulated aspart of the model. Each of the weights relays the influence of theirrespective independent variables on the dependent economic variable. Oneobjective function represents an estimation error between the dependenteconomic variable and a predicted dependent economic variable at eachtime interval. The predicted dependent economic variable is determinedat each time interval as a function of the weight of each independentvariable and its respective independent variable. The model includes oneor more other objective functions representing a transition error ofeach weight between time intervals.

Another embodiment of the present invention describes a method ofestimating time-varying weights (for example, factor exposures) forindependent variables (such as, e.g., factors or indexes) at each timeinterval in a period of time, through a dynamic optimization of a modelrelating an influence of one or more independent variables on adependent financial variable. The method includes the steps of:receiving data related to the dependent financial variable for theplurality of time intervals, receiving data related to the at least oneindependent variable for the plurality of time intervals, anddetermining at least one weight for its respective independent variableat each time interval, that minimizes values of two or more objectivefunctions while meeting at least one constraint on possible values forthe weight. The constraints and objective functions are formulated aspart of the model. Each of the weights relays the influence of theirrespective independent variables on the dependent financial variable.One objective function represents an estimation error between thedependent financial variable and a predicted dependent financialvariable at each time interval. The predicted dependent financialvariable is determined at each time interval as a function of the weightof each independent variable and its respective independent variable.The model includes one or more other objective functions representing atransition error of each weight between time intervals.

In another additional embodiment of the present invention, a method ofestimating time-varying factor exposures at each time interval in aperiod of time, through a dynamic optimization of a model relating aninfluence of at least one factor on a return of an asset collection,includes the steps of: receiving data related to the return of the assetcollection for the plurality of time intervals, receiving data relatedto the at least one factor for the plurality of time intervals; anddetermining one or more factor exposures for their respective factors ateach of the time intervals, that minimizes a value of an objectivefunction while meeting at least one constraint on possible values forone or more of the factor exposures. The asset collection includes oneor more assets, and can be, for example, a single security or aportfolio of securities, such as a mutual fund. Each of the factorexposures relays the influence of its respective factor (e.g., a returnor price of a security, or a financial or economic index) on the returnof the asset collection. The objective function includes an estimationerror term, and one or more transition error terms. The estimation errorterm represents an estimation error at each time interval between thereturn of the asset collection and a sum of products of each factorexposure and its respective factor. Each transition error termrepresents a transition error at each time interval after a first timeinterval for each factor exposure between the time interval and a priortime interval.

In another additional embodiment of the present invention, a method ofestimating time-varying factor exposures at each time interval in aperiod of time, through a dynamic optimization of a model relating aninfluence of at least one factor on a return of an asset collection,includes the steps of: receiving data related to the return of the assetcollection for all of the time intervals, receiving data related to eachfactor for all of the time intervals, and determine one or more factorexposures at each time interval, that minimizes a value of an objectivefunction while meeting one or more constraints on possible values forone or more factor exposure. The asset collection includes one or moreassets, and can be, for example, a single security or a portfolio ofsecurities, such as a mutual fund. Each of the factor exposures relaysthe influence of its respective factor (e.g., a return or price of asecurity, or a financial or economic index) on the return of the assetcollection. The objective function includes an estimation error term andone or more transition error terms. The estimation error term representsan estimation error at each time interval between the return of theasset collection and a sum of products of each factor exposure and itsrespective factor. Each transition error term represents a transitionerror at each time interval before a last time interval, for each of theat least one factor exposure between the time interval and a subsequenttime interval.

Another additional embodiment of the present invention sets forth amethod for determining structural breakpoints for one or more factorsinfluencing the return of an asset collection over a period of time madeup of a plurality of time intervals. As described more fully below, thestructural breakpoint can identify a change, for example, in themanagement of a mutual fund, or a reaction to a sudden event, e.g., anunscheduled announcement by the Federal Reserve that it is loweringinterest rates. The asset collection includes one or more assets. Theasset collection can refer to, for example, a single security, or aportfolio of securities, such as a mutual fund. The method includes thesteps of: receiving data related to the return of the asset collectionfor the plurality of time intervals, receiving data related to eachfactor for the plurality of time intervals, determining one or morefactor exposures for each of their respective factors at each timeinterval, that minimizes a value of a function including an estimationerror term and one or more transition error terms, and determiningstructural breakpoint ratios for one or more factor exposures in orderto determine their structural breakpoints. Each factor exposure relaysthe influence of a respective factor on the return of the assetcollection. The minimized function can be formulated as a parameterweighted sum. The parameter-weighted sum is a sum of a quadratic norm ofthe estimation error term and one or more parameter-weighted quadraticnorms of transition error terms. The estimation error term represents anestimation error at each time interval between the return of the assetcollection and a sum of products of each of the at least one factorexposure and its respective factor. Each transition error termrepresents transition error at each time interval after a first timeinterval for each factor exposure between the time interval and a priortime interval. The structural breakpoint ratio is determined at eachtime interval, as a ratio of a minimum of the parameter-weighted sumover all the time intervals to a minimum of a modifiedparameter-weighted sum. The modified parameter-weighted sum can beformulated as a sum of a quadratic norm of the estimation error term anda parameter-weighted quadratic norm of the at least one transition errorterm over all the time intervals. The modified parameter-weighted sumexcludes at least part of a transition error term representing atransition error of the factor exposure between the time interval and aprior time interval.

Another additional embodiment of the present invention sets forth amethod of configuring a model relating factor exposures for each of oneor more factors and their respective factors on a return of an assetcollection over a period of time. The period of time includes aplurality of time intervals. The asset collection includes one or moreassets, and as set forth above, can be a single security or a portfolio.The method includes the steps of: receiving data related to an actualreturn of the asset collection for the plurality of time intervals,receiving data related to each factor for the plurality of timeintervals, creating a reduced dataset for each particular time interval(identified as a tested time interval) excluding the data related to theactual return of the asset collection at the tested time interval, anddetermining with each reduced dataset, for each tested time interval, apredicted return of the asset collection as a function of a set ofpredicted factor exposures and their respective factors, and determininga cross validation statistic over the period of time, as a function of adifference between the actual return of the asset collection and thepredicted return of the asset collection at each time interval. Thepredicted factor exposures relay a predicted influence of a respectivefactor on the actual return of the asset collection. The set ofpredicted factor exposures for each tested time interval, are determinedwith the reduced dataset, by determining, at each time interval, thepredicted factor exposures minimizing a sum of a quadratic norm of anestimation error term and at least one parameter-weighted transitionerror term. The estimation error term, for a set of predicted factorexposures at a tested time interval, represents an estimation error ateach time interval, except for the tested time interval, between theactual return of the asset collection and a sum of products of eachpredicted factor exposure and its respective factor. Each of thetransition error terms, for a set of predicted factor exposures at atested time interval, represents a transition error at each timeinterval after a first time interval for each of the at least one factorexposure between the time interval and a prior time interval.

Another additional embodiment of the present invention sets forth amethod for evaluating a performance of an asset collection over a periodof time, for example, for evaluating the management style of a fundmanager. The asset collection includes one or more assets, as set forthabove in other embodiments. The period of time includes a plurality oftime intervals. The method includes the step of receiving informationrelating an influence of each factor in a set including at least onefactor on a return of the asset collection at each time interval. Theinfluence of each factor is determined as a function of the factorexposures of each respective factor. One of more of the factor exposuresdetermined at each time interval minimizes a value of an objectivefunction while meeting at least one constraint on possible values forone or more of the factor exposures. The objective function includes anestimation error term and one or more transition error terms. Theestimation error term represents an estimation error at each timeinterval between the return of the asset collection and a sum ofproducts of each of the at least one factor exposure and its respectivefactor. Each transition error term represents a transition error at eachtime interval after a first time interval in the period of time, foreach factor exposure between the time interval and a prior timeinterval.

Another additional embodiment of the present invention sets forth amethod for evaluating a performance of an asset collection over a periodof time. The asset collection includes one or more assets. The period oftime includes a plurality of time intervals. The method includes thestep of providing information relating an investment style of a managerof the asset collection as a function of information relating aninfluence of each factor in a set including at least one factor on areturn of the asset collection at each time interval. The influence ofeach factor is determined as a function of the factor exposures of eachrespective factor, at least one factor exposure being determined at eachtime interval that minimizes a value of an objective function whilemeeting at least one constraint on possible values for the factorexposures. The objective function includes an estimation error term andone or more transition error terms. The estimation error term representsan estimation error at each time interval between the return of theasset collection and a sum of products of each factor exposure and itsrespective factor. Each transition error term represents a transitionerror at each time interval after a first time interval in the period oftime, for each factor exposure between the time interval and a priortime interval.

Another additional embodiment sets forth a computer system forestimating time-varying factor exposures at each time interval in aperiod of time, through a dynamic optimization of a model relating aninfluence of at least one factor on a return of an asset collection. Thecomputer system includes one or more processors configured to: receivedata related to the return of the asset collection for the plurality oftime intervals, the asset collection including at least one asset;receive data related to the at least one factor for the plurality oftime intervals, and for each of the plurality of time intervals,determine at least one factor exposure minimizing a value of anobjective function while meeting at least one constraint on possiblevalues for the at least one factor exposure. Each of the at least onefactor exposures relays the influence of a respective factor on thereturn of the asset collection. The objective function includes anestimation error term and one or more transition error terms. Theestimation error term represents an estimation error at each timeinterval between the return of the asset collection and a sum ofproducts of each of the at least one factor exposure and its respectivefactor. Each transition error term represents a transition error at eachtime interval after a first time interval for each of the at least onefactor exposure between the time interval and a prior time interval.

Another additional embodiment sets forth a computer system forevaluating a performance of an asset collection over a period of time.The asset collection includes one or more assets. The period of timeincludes a plurality of time intervals. The computer system includes atleast one processor configured to receive information relating aninvestment style of a manager of the asset collection as a function ofan influence of each factor in a set including at least one factor on areturn of the asset collection at each time interval. The influence ofeach factor is determined as a function of the factor exposures of eachrespective factor. One or more of the factor exposures are determined ateach time interval, that minimize a value of an objective function whilemeeting at least one constraint on possible values for one or more ofthe factor exposures. The objective function includes an estimationerror term and one or more transition error terms. The estimation errorterm represents an estimation error at each time interval between thereturn of the asset collection and a sum of products of each of the atleast one factor exposure and its respective factor. Each of thetransition error terms represent a transition error at each timeinterval after a first time interval in the period of time, for eachfactor exposure between the time interval and a prior time interval.

Another additional embodiment of the present invention sets forth acomputer program product for use with a system evaluating a performanceof an asset collection. The computer program product includes computerusable medium having computer readable program code embodied in themedium for causing a computer to: receive data related to the return ofthe asset collection for the plurality of time intervals, receive datarelated to each factor for the plurality of time intervals, and for eachof the plurality of time intervals, determine at least one factorexposure minimizing a value of an objective function while meeting atleast one constraint on possible values for the factor exposures. Theasset collection includes one or more assets, as set forth above. Eachfactor exposure relays the influence of its respective factor on thereturn of the asset collection. The objective function includes anestimation error term and one or more transition error terms. Theestimation error term represents an estimation error at each timeinterval between the return of the asset collection and a sum ofproducts of each of the at least one factor exposure and its respectivefactor: Each transition error term represents a transition error at eachtime interval after a first time interval for each of the at least onefactor exposure between the time interval and a prior time interval.

An additional embodiment of the present invention sets forth an articleof manufacture including an information storage medium encoded with acomputer-readable data structure adapted for use in evaluating over theInternet a performance of an asset collection. The data structureincludes: one or more data fields with information related to a returnof an asset collection for a plurality of time intervals in a period oftime; one or more data fields with information related to each factorfor the plurality of time intervals, each factor influencing the returnof the asset collection; and at least one data field with informationrelated to one or more factor exposures for the plurality of timeintervals. Each factor exposure relays the influence of a respectivefactor on the return of the asset collection. The asset collectionincludes one or more assets. Each factor exposure at each time intervalis determined by minimizing a value of an objective function whilemeeting at least one constraint on possible values for the at least onefactor exposure. The objective function includes an estimation errorterm and one or more transition error terms. The estimation error termrepresents the estimation error at each time interval between the returnof the asset collection and a sum of products of each of the at leastone factor exposure and its respective factor. Each transition errorterm represents a transition error at each time interval after a firsttime interval for each of the at least one factor exposure between thetime interval and a prior time interval.

Another additional embodiment of the present invention sets forth anarticle of manufacture including a propagated signal adapted for use ina method of estimating time-varying factor exposures at each timeinterval in a period of time, through a dynamic optimization of a modelrelating an influence of at least one factor on a return of an assetcollection. The method includes the steps of: receiving data related tothe return of the asset collection for the plurality of time intervals,the asset collection including at least one asset; receiving datarelated to the at least one factor for the plurality of time intervals;and for each of the plurality of time intervals; and determining atleast one factor exposure minimizing a value of an objective functionwhile meeting at least one constraint on possible values for the atleast one factor exposure. The signal is encoded with machine-readableinformation relating to the asset collection. Each factor exposurerelays the influence of a respective factor on the return of the assetcollection. The objective function includes an estimation error term andone or more transition error terms. The estimation error term representsan estimation error at each time interval between the return of theasset collection and a sum of products of each of the at least onefactor exposure and its respective factor. Each transition error termrepresents a transition error at each time interval after a first timeinterval for each of the at least one factor exposure between the timeinterval and a prior time interval.

DETAILED DESCRIPTION 1 A Solution for Dynamic Multi-Factor Problems inFinance

The present invention is described in relation to systems and methodsfor the resolution of the dynamic multi-factor RBSA problem in finance,but can be applied to any dynamic multi-factor financial or economicproblem, in order to estimate the time-varying weights or factorexposures that model the behavior of any dependent financial or economicvariable with independent variables over a period of time. In thisdetailed description, the dependent financial variable is the return ofa single security or instrument, or the return of a portfolio ofsecurities or instruments, or any function thereof.

However, in other embodiments, the dependent financial variable can be,for example, the price of a financial instrument or portfolio, afunction of the price or return of the instrument or portfolio, or afunction including a logarithm of the price or return of the instrumentor portfolio. The independent variables may be any type of factor orindex. The factors can be the prices or returns (or functions of pricesor returns) of securities or classes of securities in a portfolio,securities or classes of securities not included in a portfolio,financial or economic indexes or other measurements that are asserted asinfluencing the behavior of the independent variable over the period oftime, or any function thereof. The factor exposures discussed below areone type of weight that relays the influence of the independentvariables on the dependent financial or economic variable in the model.The constraints set forth below can apply to one or more of theindependent variables, as part of the model that is subject to thedynamic optimization process. The labels of variables in thisapplication as dependent and independent variables are used forillustrative purposes only, in order to describe inputs to the model orproblem, and do not imply or impart any statistical dependence orindependence between any of these inputs.

1.1 The General Principle of Estimating Dynamic Multi-Factor Models

One embodiment sets forth the general principle to estimate time-varyingfactor exposures of either an individual financial instrument or aportfolio of such instruments described in equations (1) to (4) above,as a method that consists in solving a constrained multi-criteriadynamic optimization problem containing m+2 objective functions to beminimized, which are associated, respectively, with the estimation errorvector and m+1 transition error vectors in certain norms:

$\begin{matrix}{\quad\left\{ \begin{matrix}{{\min\limits_{\underset{({\beta_{1},\ldots,\beta_{N}})}{({\alpha_{1},\ldots,\alpha_{N}})}}{e_{\lbrack{1,\ldots,N}\rbrack}}},{\min\limits_{({\alpha_{1},\ldots,\alpha_{N}})}{w_{{0,{\lbrack{2,\ldots,N}\rbrack}}\mspace{14mu}}}},{\min\limits_{({\beta_{1},\ldots,\beta_{N}})}{w_{1,{\lbrack{{k_{1} + 1},\ldots,N}\rbrack}}}},\ldots\mspace{11mu},{\min\limits_{({\beta_{1},\ldots,\beta_{N}})}{w_{m,{\lbrack{{k_{m} + 1},\ldots,N}\rbrack}}}},} \\{{e_{\lbrack{1,\ldots\mspace{11mu},N}\rbrack} = {\left( {{e_{t} = {y_{t} - \alpha_{t} - {x_{t}^{T}\beta_{t}}}},{t = 1},\ldots\mspace{11mu},N} \right)\mspace{14mu}\left( {N\text{-}{dimensional}\mspace{14mu}{vector}\mspace{14mu}{of}\mspace{14mu}{estimation}\mspace{14mu}{errors}} \right)}},} \\{\begin{matrix}{{\left. \begin{matrix}{w_{0,{\lbrack{2,\ldots,N}\rbrack}} = \left( {{w_{0,t} = {{\alpha_{t} - \alpha_{t - 1}}}},{t = 1},...\mspace{11mu},N} \right)} \\{{w_{j,{\lbrack{{k_{j} + 1},\ldots,N}\rbrack}} = \left( {{w_{j,t} = {{\beta_{t} - {\overset{¨}{\beta}}_{t}}}},{t = {k_{j} + 1}},\ldots\mspace{11mu},N} \right)},} \\{{j = 1},\ldots\mspace{11mu},m}\end{matrix} \right\}\left( {N\text{-}{dimensional}\mspace{14mu}{vectors}\mspace{14mu}{of}\mspace{14mu}{transition}\mspace{14mu}{errors}} \right)},} \\{{{\overset{¨}{\beta}}_{j,t} = {{f_{j}\left( {x_{t},\ldots\mspace{11mu},{x_{t - k_{j}};\beta_{t - 1}},\ldots\mspace{11mu},\beta_{t - k_{j}}} \right)}\mspace{14mu}\left( {{transition}\mspace{14mu}{equations}} \right)}},} \\{{e_{\lbrack{1,\ldots,N}\rbrack}},{{w_{j,{\lbrack{{k_{j} + 1},\ldots,N}\rbrack}}}\mspace{14mu}{and}\mspace{14mu}{{\beta_{t}^{(i)} - {\overset{¨}{\beta}}_{t}^{(i)}}}\mspace{14mu}{are}\mspace{14mu}{norms}\mspace{14mu}{of}\mspace{14mu}{any}\mspace{14mu}{kinds}},}\end{matrix}\begin{matrix}\mspace{14mu} \\\mspace{14mu}\end{matrix}}\end{matrix} \right.} & {{Equation}\mspace{14mu}(10)}\end{matrix}$subject to constraints:

$\begin{matrix}\left\{ \begin{matrix}{{{G_{t}{\overset{\_}{\beta}}_{t}} + b_{t}} \geq 0} & {\left( {{inequality}\mspace{14mu}{constraints}} \right),} \\{{{F_{t}{\overset{\_}{\beta}}_{t}} + c_{t}} = 0} & {\left( {{equality}\mspace{14mu}{constraints}} \right),{{\overset{\_}{\beta}}_{t} = {\left( {\alpha_{t},\beta_{t}} \right).}}}\end{matrix} \right. & (11)\end{matrix}$

-   Here: y_(t) present the given return (performance) of an instrument    portfolio during period or time interval t, or any transformation of    them, for instance, logarithmic;    -   β_(t)=(β_(t) ⁽¹⁾, . . . , β_(t) ^((n)))^(T) are unknown        n-dimensional column vectors of factor exposures during period        t;    -   α_(t) is the unknown scalar intercept term during period t;    -   β _(t)=(α_(t),β_(t) ^(T))^(T)=(α_(t),β_(t) ⁽¹⁾, . . . , β_(t)        ^((n)))^(T) are extended (n+1)-dimensional vectors consisting of        the factor exposures and intercept term.    -   x_(t)=(x_(t) ⁽¹⁾, . . . , x_(t) ^((n)))^(T) are known        n-dimensional column vectors during period t; in the RBSA model        these are returns on generic indices, while in the APT models        these represent changes in certain economic factors;    -   {umlaut over (β)}_(j,t)=f_(j)(x_(t), . . . , x_(t−k) _(j) ;        β_(t−1), . . . , β_(t−k) _(j) ) are n-dimensional column vectors    -   {umlaut over (β)}_(j,t)=(β_(j,t) ⁽¹⁾, . . . , β_(j−t)        ^((n)))^(T) of transitioned factor exposures during period t in        accordance with a linear or nonlinear model of the hidden        dynamics of time-varying factor exposures; each criterion j=1, .        . . , m corresponds to a specific assumption on the dynamics        model expressed by the choice of the respective function        f_(j)(x_(t), . . . , x_(t−k) _(j) ;β_(t−1), . . . , β_(t−k) _(j)        ).    -   G_(t) β _(t)+h_(t)≧0, G_(t)[l×(n+1)], h_(t)(l), are l time        varying inequality constraints that represent prior information        about coefficients of the model, for example non-negativity of        exposures in the style analysis model (4) or hedging constraints        in (20) further in the text;    -   F_(t) β _(t)+c_(t)0, F_(t)[p×(n+1)], c_(t)(p), are p general        linear equality constraints that represent certain knowledge        about the structure of the parameters, for example, the budget        constraint

${\sum\limits_{i = 1}^{n}\beta_{t}^{(i)}} = 1$in the style analysis model (4).1.2 Optimization Problem Formulation.

The model presented in Section 1.1 above, can be formulated as set forthbelow.

1.2.1. Criteria in the General Multi-Objective Principle of EstimatingTime-Varying Factor Exposures.

In this embodiment, the elementary criteria in the generalmulti-objective principle of estimating time-varying factor exposuresfrom (10), can be expressed as follows:

-   -   1) The squared Euclidean norm for estimation error vector        e_([1, . . . , N])=(e₁, . . . , e_(N)) representing the errors        of fit

$\begin{matrix}{{{e_{\lbrack{1,\ldots,N}\rbrack}} = {\sum\limits_{t = 1}^{N}e_{t}^{2}}},{e_{t} = {y_{t} - \alpha_{t} - {x_{t}^{T}{\beta_{t}.}}}}} & (12)\end{matrix}$

-   -   2) The squared Euclidean norms for transition error vectors of        the intercept term        w _(0,[2, . . . , N])=(w _(0,1) , . . . , w _(0,N)),w        _(0,t)=(α_(t)−α_(t−1))²,        and factor exposures w_(j,[k) _(j) _(+1, . . . , N])=(w_(j,k)        _(j) ₊₁, . . . , w_(j,N))

$\begin{matrix}{{{w_{j,{\lbrack{{k_{j} + 1},\ldots,N}\rbrack}}}^{2} = {\sum\limits_{t = {k_{j} + 1}}^{N}w_{j,t}^{2}}},{w_{j,t} = \left\lbrack {\left( {\beta_{t} - {\overset{¨}{\beta}}_{t}} \right)^{T}{U_{j,t}\left( {\beta_{t} - {\overset{¨}{\beta}}_{t}} \right)}} \right\rbrack^{1/2}}} & (13)\end{matrix}$

-   -   between the actual values of factor exposures β_(t) and        transitioned values        {umlaut over (β)}_(t) =V _(j,1)β_(t−1) + . . . +V _(j,k) _(j)        β_(t−k) _(j)   (14)    -   in accordance with a linear model of the dynamics of        time-varying factor exposures; each criterion j=1, . . . , m        corresponds to a specific assumption on the linear model defined        by the choice of transition matrices V_(j,1), . . . , V_(j,k)        _(j) ; the positive semi-definite matrices U_(j,t) (n×n) are        defined to a) provide a proper unit scaling between the        transition error

$\begin{matrix}{w_{j,t} = \left\lbrack {\sum\limits_{i = 1}^{n}{\sum\limits_{l = 1}^{n}{{u_{j,t}^{({il})}\left( {\beta_{t}^{(i)} - {\overset{¨}{\beta}}_{j,t}^{(i)}} \right)}\left( {\beta_{t}^{(l)} - {\overset{¨}{\beta}}_{j,t}^{(l)}} \right)}}} \right\rbrack^{1/2}} & (13)\end{matrix}$and the fit error e_(t) (12), and b) to individually (per factor) adjusttransition errors. Unit scaling is desirable in multi-criteriaoptimizations because it provides common measurement units for allcriteria. In our multi-criteria model, the transition errors areproportional to squared exposure deviations, while the fit errors arealso proportional to squared factor changes. However, other norms (e.g.,deviations or squared deviations) and other scaling can be used inmulti-criteria models for the transition errors and fit errors inadditional embodiments of the present invention.

-   -   In most cases it is sufficient to define matrix U=diag(X^(T)X),        where X is the N×n matrix of N factor raw-vectors x_(t)        ^(T)=(x_(t) ⁽¹⁾, . . . , x_(t) ^((n))).    -   The number of transition equations m depends on the amount of a        priori information known about the financial instrument (or        portfolio) y that is being analyzed. For example, the        requirement of exposure paths to be smooth and at the same time        at the end of each quarter to revert to a certain (same) value        may result in two transition criteria.        1.2.2. General Quadratic Optimization Problem.

In one embodiment, a method of multi-criteria estimating factorexposures for Euclidean norms of fit (12) and transition errors (13)under the linear model of the dynamics of factor exposures (14) includessolving the following quadratic programming problem, i.e. a quadraticoptimization problem under linear equality and inequality constraints,formed as a linear combination of elementary quadratic criteria into acombined quadratic criterion with m free dimensionless coefficientsλ₀≧0, λ₁≧0, . . . , λ_(m)≧0 under constraints (11):

$\begin{matrix}\left\{ \begin{matrix}{\min\limits_{\underset{\beta_{1},\ldots,\beta_{N}}{\alpha_{1},\ldots,\alpha_{N}}}\left\lbrack {{\sum\limits_{t = 1}^{N}\left( {y_{t} - \alpha_{t} - {x_{t}^{T}\beta_{t}}} \right)^{2}} + {\lambda_{0}{\sum\limits_{t = 2}^{N}\left( {\alpha_{t} - \alpha_{t - 1}} \right)^{2}}} + {\lambda_{1}{\sum\limits_{t = {k_{1} + 1}}^{N}{\left( {\beta_{t} - {\overset{¨}{\beta}}_{1,t}} \right)^{T}{U_{1,t}\left( {\beta_{t} - {\overset{¨}{\beta}}_{1,t}} \right)}}}} + \ldots +} \right.} \\{\left. {{\lambda_{m}{\sum\limits_{t = {k_{m} + 1}}^{N}{\left( {\beta_{t} - {\overset{¨}{\beta}}_{m,t}} \right)^{T}U_{m,t}}}} - \left( {\beta_{t} - {\overset{¨}{\beta}}_{m,t}} \right)} \right\rbrack,{{\overset{¨}{\beta}}_{j,t} = {{V_{j,1}\beta_{t - 1}} + \ldots + V_{j,k_{j}}}},{{\beta_{{t - k_{j}},}j} = 1},\ldots\mspace{11mu},m,} \\{{{{{subject}\mspace{14mu}{to}\mspace{14mu} G_{t}{\overset{\_}{\beta}}_{t}} + h_{t}} \geq 0},{{{F_{t}{\overset{\_}{\beta}}_{t}} + c_{t}} = 0},{t = 1},\ldots\mspace{11mu},{N.}}\end{matrix} \right. & (15)\end{matrix}$

The objective function is quadratic min_({tilde over (β)})[{tilde over(β)}^(T){tilde over (Q)}{tilde over (β)}+{tilde over (q)}^(T){tilde over(β)}+b] with respect to the combined (n+1)N-dimensional variable {tildeover (β)}=( β ₁ ^(T), . . . , β _(N) ^(T))^(T), where matrix {tilde over(Q)} [(n+1)N×(n+1)N] and vector {tilde over (q)} (n+1)N are built fromblocks that depend on the given time series (y_(t), x_(t)), t=1, . . . ,N, and the parameters of the method, namely, matrices U_(j,s), V_(j,t),j=1, . . . , m, t=1, . . . , N, and weighting coefficients λ_(j), j=0,1, . . . , m. Analogously, the N inequality constraints and N equalityones in (15) will be expressed, with respect to the combined variable{tilde over (β)}=( β ₁ ^(T), . . . , β _(N) ^(T))^(T), by two equivalentconstraints {tilde over (G)}{tilde over (β)}+{tilde over (h)}≧0, {tildeover (F)}{tilde over (β)}+{tilde over (c)}=0, where matrices {tilde over(G)} [l×(n+1)N], {tilde over (F)} [p×(n+1)N], and vectors {tilde over(h)} (l), {tilde over (c)} (p) consist of blocks, respectively,G_(t)[l×(n+1)], F_(t)[p×(n+1)], h_(t)(l) and c_(t)(p).

Such a quadratic problem

$\quad\left\{ \begin{matrix}\begin{matrix}\begin{matrix}{\min_{\overset{\sim}{\beta}}\left\lbrack {{{\overset{\sim}{\beta}}^{T}\overset{\sim}{Q}\;\overset{\sim}{\beta}} + {{\overset{\sim}{q}}^{T}\overset{\sim}{\beta}} + b} \right\rbrack} \\{{subject}\mspace{14mu}{to}}\end{matrix} \\{{{{\overset{\sim}{G}\;\overset{\sim}{\beta}} + \overset{\sim}{h}} \geq 0},}\end{matrix} \\{{{\overset{\sim}{F}\;\overset{\sim}{\beta}} + \overset{\sim}{c}} = 0}\end{matrix} \right.$can be solved by any standard quadratic programming procedure (aprocedure of quadratic optimization under linear equality and inequalityconstraints), for instance, based on an interior point method which isvery efficient in solving large problems (described, for example, inWright, S., Primal-dual interior-point methods. SIAM, 1997).1.2.3. Specific Quadratic Optimization Problem.

In another embodiment, a method of estimating the time-varying factorexposures in the constrained dynamic RBSA problem (7) includes solvingthe following quadratic programming problem:

$\begin{matrix}\left\{ \begin{matrix}{{\min\limits_{\underset{\beta_{1},\ldots,\beta_{N}}{\alpha_{1},\ldots,\alpha_{N}}}\left\lbrack {{\sum\limits_{t = 1}^{N}\left( {y_{t} - \alpha_{t} - {x_{t}^{T}\beta_{t}}} \right)^{2}} + {\lambda_{0}\left( {\alpha_{t} - \alpha_{t - 1}} \right)}^{2} + {\lambda_{1}{\sum\limits_{t = 2}^{N}{\left( {\beta_{t} - {V_{t}\beta_{t - 1}}} \right)^{T}{U_{t}\left( {\beta_{t} - {V_{t}\beta_{t - 1}}} \right)}}}}} \right\rbrack},} \\{{\lambda_{0} > 0},{\lambda_{1} > 0},} \\{{{{subject}\mspace{14mu}{to}\mspace{14mu}{\sum\limits_{i = 1}^{n}\beta_{t}^{(i)}}} = {1\mspace{14mu}\left( {{budget}\mspace{14mu}{constraint}} \right)}},{\beta_{t}^{(i)} \geq {0\mspace{14mu}{\left( {{nonnegativity}\mspace{14mu}{bounds}} \right).}}}}\end{matrix} \right. & (16)\end{matrix}$

This formulation of the problem is a special case of (15) and, thesolution can processed by any standard quadratic programming procedure.

The coefficients λ₀ and λ₁ are responsible for smoothness of,respectively, the intercept term α_(t) and factor exposures β_(t). Thelarger the values of these coefficients, the more weight is attached tothe respective penalty term, the smoother the solution. For example, ifλ₀→∞ and λ₁→∞, the solution of (16) becomes very smooth and approaches aleast squares solution over the entire range of observations, whileselecting λ₀ and λ₁ close to zero makes the paths α_(t) and β_(t)=(β_(t)⁽¹⁾, . . . , β_(t) ^((n)))^(T) very volatile.

Note that the role of each λ in (16) is similar to the role of themoving window size described in the “BACKGROUND OF THE INVENTION”section above, namely, the wider the moving window, the smoother are theobtained paths of the respective time varying model parameter. Theextreme case is when the window coincides with the whole date range,when both methods produce the same least squares solution (5).

The sequence of positive semi-definite matrices (U₂, . . . , U_(N))plays the role of additional free parameters of the data model thatprovide a) a priori information about the relative smoothness of factorexposure vectors β_(t)=(β_(t) ⁽¹⁾, . . . , β_(t) ^((n))) as a whole ateach pair of adjacent time moments (t−1,t) the difference of therequired smoothness of individual exposure coefficients β_(t) ^((i)).

In most cases, it can be enough to choose diagonal matricesU_(t)=Diag(u_(t) ^((i))≧0, i=1, . . . , n) as defined in section 1.2.1.The greater u_(t) ^((i)), the more smooth the sought-for coefficientβ_(t) ^((i)) is assumed to be at (t−1, t). For example, setting valueu_(t) ^((i))=0 is equivalent to the assumption that β_(t) ^((i))undergoes a structural break at this point of time.

1.2.4. Nonlinear Transition Terms.

When factors in the model (10) represent financial assets (financialinstruments, market indices, etc.) the transition errors (13)w_(j,[2, . . . , N])=(w_(j,1), . . . , w_(j,N)) include changes inexposures to these factors induced by the market. For example, if marketvalue of a factor index in the model (10) changes dramatically during asingle time period as compared to other model factors, then the relativevalue of exposure to this index has to undergo a similar change, and,therefore, non-smoothness of the exposure in this time period is notonly normal, but is essential. At the same time, quadratic penalty termin model (15) corresponding to the transition errors on this factor issmoothing the exposure path. A way to remove such factor-induced driftfrom the transition errors is described below.

The following nonlinear transition equation is applied to (15):f(x _(t−1) , . . . , x _(t−k);β_(t−1), . . . , β_(t−k))=V(β_(t−1) ,x_(t−1))x _(t−1)  (17)where the variable- and data-dependent matrix V(β_(t−1),x_(t−1)) (n×n)is diagonal

$\begin{matrix}{{{V\left( {\beta_{t - 1},x_{t - 1}} \right)} = {{Diag}\left\lbrack {{v_{t}^{(i)}\left( {\beta_{t - 1},x_{t - 1}} \right)},{i = 1},\ldots\mspace{11mu},n} \right\rbrack}},{{v_{t}^{(i)}\left( {\beta_{t - 1},x_{t - 1}} \right)} = {\frac{1 + x_{t - 1}^{(i)}}{\sum\limits_{l = 1}^{n}{\beta_{t - 1}^{(l)}\left( {1 + x_{t - 1}^{(l)}} \right)}}.}}} & (18)\end{matrix}$

In Equation (18), the term on the right-hand side of the last equalityrepresents the relative weight of each of the index/factor exposures inthe effective mix induced by the change in corresponding index/factorover period t. Note that the scaling factor in denominator is requiredto satisfy the budget constraint

${\sum\limits_{l = 1}^{n}{\beta_{t - 1}^{(l)}\left( {1 + x_{t - 1}^{(l)}} \right)}} = {1 + {\sum\limits_{l = 1}^{n}{\beta_{t - 1}^{(l)}{x_{t - 1}^{(l)}.}}}}$As a result of the presence of nonlinear transition equation (17), theobjective function is no longer quadratic, and the optimization problemis no longer a quadratic programming problem. For example, the objectivefunction in the problem (15) with m=1 and f₁(x_(t−1), . . . , x_(t−k);β_(t−1), . . . , β_(t−k))=V(β_(t−1),x_(t−1))x_(t−1) will be transformedas follows:

$\begin{matrix}{{\min\limits_{\underset{\beta_{1},\ldots,\beta_{N}}{\alpha_{1},\ldots,\alpha_{N}}}\left\lbrack {{\sum\limits_{t = 1}^{N}\left( {y_{t} - \alpha_{t} - {x_{t}^{T}\beta_{t}}} \right)^{2}} + {\lambda_{0}{\sum\limits_{t = 2}^{N}\left( {\alpha_{t} - \alpha_{t - 1}} \right)^{2}}} + {\lambda_{1}{\sum\limits_{t = 2}^{N}{\left( {\beta_{t} - {{V\left( {\beta_{t - 1},x_{t - 1}} \right)}\beta_{t - 1}}} \right)^{T}{U_{t}\left( {\beta_{t} - {{V\left( {\beta_{t - 1},x_{t - 1}} \right)}\beta_{t - 1}}} \right)}}}}} \right\rbrack},\mspace{20mu}{{V\left( {\beta_{t - 1},x_{t - 1}} \right)} = {{{Diag}\left( {\frac{1 + x_{t - 1}^{(i)}}{1 + {x_{t - 1}^{T}\beta_{t - 1}}},{i = 1},\ldots\mspace{11mu},n} \right)}.}}} & (19)\end{matrix}$1.3 New Constraints for the RBSA Model.

In one embodiment, the RBSA method (16) can be extended by suggestingthe following general constraints which account for more complex apriori information about portfolio/instrument structure (for example,short-selling, hedging, leveraging):

$\begin{matrix}{{{\sum\limits_{i \in \Omega}^{\;}\beta_{t}^{(i)}} = {1\mspace{14mu}\left( {{{optional}\mspace{14mu}{budget}\mspace{14mu}{constraint}\mspace{14mu}{for}\mspace{14mu} a\mspace{14mu}{subset}\text{}{of}\mspace{14mu}{the}\mspace{14mu}{full}\mspace{14mu}{set}\mspace{14mu}{of}\mspace{14mu}{factors}\mspace{14mu}\Omega} \Subset \left\{ {1,\ldots\mspace{11mu},n} \right\}} \right)}},{g_{t}^{(i)} \leq \beta_{t}^{(i)} \leq {h_{t}^{(i)}\mspace{14mu}\left( {{optional}\mspace{14mu}{individual}\mspace{14mu}{bounds}} \right)}},{{{- H_{t}}{\sum\limits_{k \in B}{\beta_{t}^{(k)}\omega_{t}^{(k)}}}} \leq {\sum\limits_{i \in A}{\beta_{t}^{(i)}\omega_{t}^{(i)}}} \leq {{- L_{t}}{\sum\limits_{k \in B}^{\;}{\beta_{t}^{(k)}\omega_{t}^{(k)}}}}},{B \Subset \left\{ {1,\ldots\mspace{11mu},n} \right\}},{A \Subset {\left\{ {1,\ldots\mspace{11mu},n} \right\}\mspace{14mu}\left( {{optional}\mspace{14mu}{hedging}} \right)}},{{\sum\limits_{i \in \Phi}{\min\left( {\beta_{t}^{(i)},0} \right)}} \geq {- S_{t}}},{\Phi \Subset {\left\{ {1,\ldots\mspace{11mu},n} \right\}\mspace{14mu}{\left( {{{optional}\mspace{14mu}{short}} - {{selling}\mspace{14mu}{leveraging}}} \right).}}}} & (20)\end{matrix}$

Here:

-   -   A budget constraint can be specified only for a subset of        |Ω|=n′<n indices, where |Ω| is the number of elements in the        respective finite set. For example, the rest of the assets could        be hedging instruments (currency forward rates).    -   The hedging constraint is interpreted as follows: a portfolio of        hedging instruments B⊂{1, . . . , n} (a subset of all assets        |B|≦n) is hedging a portfolio A⊂{1, . . . , n}, |A|≦n, with a        hedge ratio within time varying interval (H_(t),L_(t)). Known        parameters ω_(t) ^((i)) define the known proportion between        assets in the hedging portfolio and hedged portfolio. Unless        these weights are known, typically they would be set to 1. In        the simplest case, for example, the hedging portfolio contains        one element—the currency return index in US dollar, and the        hedged portfolio represents several generic indices        (fixed-income, equity, etc.) in US dollar. Note that there could        be several hedging constraints depending on the number of        hedging relationships in the overall portfolio.        1.4 Numerical Solution.

Once the dynamic optimization problem is formulated, as set forth, forexample, in Equation (15), the numerical solution can be obtained byprocessing the optimization problem in a variety of ways, as set forthbelow.

1.4.1. General Method to Solve an RBSA Problem (15).

In one embodiment, one general method of solving the RBSA problem, asformulated in Equation (15), includes programming a processor to use aquadratic programming algorithm in order to solve the problem.

1.4.2. Specific Method to Solve an RBSA Problem (15).

In another embodiment, the processor can be programmed with an interiorpoint algorithm (e.g., the algorithm described in Wright, S.,Primal-dual interior-point methods, SIAM, 1997), in order to solve theRBSA problem, as formulated in Equation (15) above.

1.4.3. Alternative Method of Solving a General Problem with Constraints(15).

In another embodiment, a processor can be programmed to implement thefollowing steps in order to solve the RBSA problem, as formulated inEquation (15) above, in a more efficient manner.

-   -   Step 1. Determine whether constraints can be dropped. For        example, a model (15) that includes only equality constraints        F_(t){umlaut over (β)}_(t)+c₁=0, in particular, the budget one

${{\sum\limits_{i = 1}^{n}\beta_{t}^{(i)}} = 1},$can be converted to an equivalent unconstrained model, like (8), butcontaining variables vectors of lesser dimensionality {dot over(β)}_(t)=(β_(t) ¹, . . . , β_(t) ^(n−p)) where p is the number ofequality constraints. For instance, the budget constraint

${\sum\limits_{i = 1}^{n}\beta_{t}^{(i)}} = 1$can be observed by setting

$\beta_{t}^{(n)} = {1 - {\sum\limits_{i = 1}^{n - 1}\beta_{t}^{(i)}}}$and solving the resulting unconstrained problem with respect to theremaining n−1 factor exposures {dot over (β)}_(t)=(β_(t) ⁽¹⁾, . . . ,β_(t) ^((n−1))).

-   -   A model without constraints with a single transition equation        for any t        β_(t) ≅Vβ _(t−1)  (21)    -   is equivalent to the FLS model (8) and can be solved by the        recursive FLS algorithm, which is an algorithm of unconstrained        quadratic optimization.    -   An unconstrained model with two transition equations        β_(t) ≅V _(1,l)β_(t−1) + . . . +V _(1,m)β_(t−m), β_(t) ≅V        ₂β_(t−s)  (22)    -   is equivalent to the GFLS model (9) and can be solved by a        recursive GFLS algorithm of unconstrained quadratic        optimization.    -   Step 2. If optimization problem (15) can be solved by a        recursive algorithm FLS or GFLS, the appropriate algorithm is        then utilized by a processor to obtain the solution. If not, a        quadratic programming solver (an algorithm of quadratic        optimization under linear equality and inequality constraints)        as mentioned above in Sections 1.2.2 and 1.2.3 is utilized by a        processor to obtain the solution.        1.4.4. Iterative Algorithm for Solution of the Problem (16)        Containing a Non-Quadratic Objective Function (19).

In one embodiment, an iterative algorithm can be used, based on the factthat the criterion (19) becomes quadratic, if the values β_(t) ^((k)) inthe denominator of the transition term are considered as predefinedconstants. A processor could be programmed with the following iterativealgorithm, to obtain a solution on step q, the solution being denoted asβ_(t)(q):

-   -   Step 0. Obtain solution β_(t)(0) of the quadratic programming        problem with objective function (16).    -   Step q>0. Use the solution obtained on the previous step as a        constant β_(t)(q−1)=const in the denominator of the scaling        factor and solve the resulting quadratic programming problem:

$\begin{pmatrix}{{\alpha_{1}(q)},\ldots\mspace{11mu},{\alpha_{N}(q)}} \\{{\beta_{1}(q)},\ldots\mspace{11mu},{\beta_{N}(q)}}\end{pmatrix} = {\underset{\underset{\beta_{1},\ldots,\beta_{N}}{\alpha_{1},\ldots,\alpha_{N}}}{\arg\;\min}\left\lbrack {{\sum\limits_{t = 1}^{N}\left( {y_{t} - \alpha_{t} - {x_{t}^{T}\beta_{t}}} \right)^{2}} + {\lambda_{0}\left( {\alpha_{t} - \alpha_{t - 1}} \right)}^{2} + {\lambda_{1}{\sum\limits_{t = 2}^{N}{\left( {\beta_{t} - {V_{t}\beta_{t - 1}}} \right)^{T}{U_{t}\left( {\beta_{t} - {V_{t}\beta_{t - 1}}} \right)}}}}} \right\rbrack}$

-   -   where V_(t) are diagonal (n×n) matrices with

$v_{ii}^{(t)} = \frac{1 + x_{t - 1}^{(i)}}{\sum\limits_{i = 1}^{n}\left\lbrack {1 + {{\beta_{t - 1}^{(i)}\left( {q - 1} \right)}x_{t - 1}^{(i)}}} \right\rbrack}$

The number of iterations can be fixed or determined by the convergenceof consecutive iterations.

2 Determining Structural Breakpoints in Factor Exposures

In one embodiment, the following method can be used to determinestructural changes in exposures in the model, presented in Equations(16)-(20).

The partial left and right objective functions for a certain point t aredenoted as follows:

${{J_{\lbrack{1,t}\rbrack}\left( {{\overset{\_}{\beta}}_{1},\ldots\mspace{11mu},{\overset{\_}{\beta}}_{t}} \right)} = {{\sum\limits_{s = 1}^{t}\left( {y_{s} - \alpha_{s} - {x_{s}^{T}\beta_{s}}} \right)^{2}} + {\lambda_{0}{\sum\limits_{s = 2}^{t}\left( {\alpha_{s} - \alpha_{s - 1}} \right)^{2}}} + {\lambda_{1}{\sum\limits_{s = 2}^{t}{\left( {\beta_{s} - {V_{s}\beta_{s - 1}}} \right)^{T}{U_{s}\left( {\beta_{s} - V_{s - 1}} \right)}}}}}},{{J_{\lbrack{t,N}\rbrack}\left( {{\overset{\_}{\beta}}_{t},\ldots\mspace{11mu},{\overset{\_}{\beta}}_{N}} \right)} = {{\sum\limits_{s = t}^{N}\left( {y_{s} - \alpha_{s} - {x_{s}^{T}\beta_{s}}} \right)^{2}} + {\lambda_{0}{\sum\limits_{s = t}^{N - 1}\left( {\alpha_{s + 1} - \alpha_{s}} \right)^{2}}} + {\lambda_{1}{\sum\limits_{s = t}^{N - 1}{\left( {\beta_{s + 1} - {V_{s + 1}\beta_{s}}} \right)^{T}{U_{s + 1}\left( {\beta_{s + 1} - {V_{s + 1}\beta_{s}}} \right)}}}}}},\mspace{79mu}{{\overset{\_}{\beta}}_{t} = {\left( {\beta_{t}^{T},\alpha_{t}} \right)^{T} = \left( {\beta_{t}^{(1)},\ldots\mspace{11mu},\beta_{t}^{(n)},\alpha_{t}} \right)^{T}}},$where the second sum in both notations is considered as equal to zeroif, respectively, t=1 and t=N−1.

The method is based on the separable property of the full objectivefunction J( β ₁, . . . , β _(N))==J_([1,N])( β ₁, . . . , β _(N)) forany t ε {2, . . . , N}:J _([1,N])( β ₁, . . . , β _(N))=J _([1,t−1])( β ₁, . . . , β _(t−1))+J_([t,N])( β _(t), . . . , β _(N))+γ_(t)( β _(t−1), . . . , β _(t)).Hereγ₁( β _(t−1), . . . , β _(t))=λ₀(α_(t)−α_(t−1))²+λ₁(β_(t) −V_(t)β_(t−1))^(T) U _(t)(β_(t) −V _(t)β_(t−1))denotes the transition term at the pair of adjacent points t−1 and t.

In this embodiment, the objective function is modified as follows:J _([1,N]) ^(t,μ)( β ₁, . . . , β _(N))=J _([1,t−1])( β ₁, . . . , β_(t−1))+J _([t,N])( β _(t), . . . , β _(N))+μγ_(t)( β _(t−1), . . . , β_(t)),where parameter 0≦<1 reflects full removal μ=0 or partial removal of thetransition term corresponding to the point t. SinceJ _([1,N]) ^(t,μ)( β ₁, . . . , β _(N))<J _([1,N])( β ₁, . . . , β_(N)),the following inequality is true:

${\min\limits_{{\overset{\_}{\beta}}_{1},\ldots,{\overset{\_}{\beta}}_{N}}{J_{\lbrack{1,N}\rbrack}^{t,\mu}\left( {{\overset{\_}{\beta}}_{1},\ldots\mspace{14mu},{\overset{\_}{\beta}}_{N}} \right)}} \leq {\min\limits_{{\overset{\_}{\beta}}_{1},\ldots,{\overset{\_}{\beta}}_{N}}{{J_{\lbrack{1,N}\rbrack}\left( {{\overset{\_}{\beta}}_{1},\ldots\mspace{14mu},{\overset{\_}{\beta}}_{N}} \right)}.}}$

The operation “min” here and below denotes the minimum of the objectivefunction subject to the optional constraints in (15).

In this embodiment, the present invention sets forth a statistic,referred to herein as a Structural Breakpoint Ratio (SBR), indicatingthe presence of structural changes in factor exposures in the model(15):

$\begin{matrix}{{\rho_{t}^{\mu} = \frac{\min\limits_{{\overset{\_}{\beta}}_{1},\ldots,{\overset{\_}{\beta}}_{N}}{J_{\lbrack{1,N}\rbrack}\left( {{\overset{\_}{\beta}}_{1},\ldots\mspace{11mu},{\overset{\_}{\beta}}_{N}} \right)}}{\min\limits_{{\overset{\_}{\beta}}_{1},\ldots,{\overset{\_}{\beta}}_{N}}{J_{\lbrack{1,N}\rbrack}^{t,\mu}\left( {{\overset{\_}{\beta}}_{1},\ldots\mspace{11mu},{\overset{\_}{\beta}}_{N}} \right)}}},{t \in {\left\{ {2,\ldots\mspace{11mu},N} \right\}.}}} & (23)\end{matrix}$

A visual or analytical analysis of SBR values can be used to determineone or several possible breakpoints. Specifically, the pair of adjacentpoints (t*−1, t*) in the interval t*ε {2, . . . , N} is considered to bethe point of a possible structural break if

$t^{*} = {\underset{t \in {\{{2,\ldots,N}\}}}{\arg\;\max}{\rho_{t}^{\mu}.}}$

In addition, a certain threshold h>1 can be used to detect a realstructural shift as follows:

$\begin{matrix}{t^{*} = {\underset{t \in {{{\{{2,\ldots,N}\}}\text{:}\rho_{t}^{\mu}} > h}}{\arg\;\max}{\rho_{t}^{\mu}.}}} & (24)\end{matrix}$

If the condition ρ_(t) ^(μ)>h is not met at any time moment, then thereare no breakpoints in the succession of factor exposures.

The above described methodology and the SBR statistic can be used forany model (15) with any transition equations that allow the objectivefunction to be separable, for example, having m=1.

3 Measuring Solution Adequacy, Determining Optimal Model Parameters

The optimization problem (15) that provides solution to the generalmodel (10,11) contains m+1 free parameters λ₀, λ₁, . . . , λ_(m) andtherefore, allows for an infinite number of solutions. Typically,researchers have presented solutions of a large number of optimizationsfor various values of parameters, where such parameters belong to an(m+1)-dimensional grid. The results, being actually the time paths forvarious values of α_(t) and β_(t), t=1, . . . , N, are then visuallyevaluated for consistency. For example, in Lütkepohl, H., Herwartz, H.,Specification of varying coefficient time series models via generalizedflexible least squares. Journal of Econometrics, 70, 1996, pp. 261-290),in the case when m=1 and the constraints are absent, the authors takethe discrete values of the parameter λ=10⁻³, λ=1, λ=10³.

Below, the present invention sets forth an embodiment including methodsto measure portfolio adequacy and automatically select the mostappropriate set of regressors (model selection) and the optimal value ofparameters λ.

3.1 Cross Validation Statistic

In this embodiment, a Cross Validation (CV) of a model is used toevaluate the ability of a model to predict (forecast). For the purposeof cross validation, the available observation data sample is typicallysplit into two sets, the estimation set and the test set. The model isthen evaluated on the former set and tested on the latter one. For thepurpose of such test, a certain statistic, a loss function, is beingcalculated based on the prediction errors in a predefined metric.

A validation statistic Q_(CV) for the general model (9) can be computedas follows:

-   -   Step 1. For each t ε {1, . . . , N}, obtain a solution        ({circumflex over ( β _(t) ^((t)), t=1, . . . , N), {circumflex        over ( β _(t) ^((t))=({circumflex over (β)}_(t) ^((t,1)), . . .        , {circumflex over (β)}_(t) ^((t,n)), {circumflex over (α)}_(t)        ^((t)))^(T), for a reduced optimization problem with estimation        equation corresponding to the return y_(t) at point t removed        {y₁, . . . , y_(t−1), y_(t+1), . . . , y_(N)}.    -   Step 2. For each t ε {1, . . . , N}, compute a prediction ŷ_(t)        ^((t)) of the removed instrument/portfolio return y_(t) as the        sum of the intercept term and the weighted average of index        (factor) returns

${\hat{y}}_{t}^{(t)} = {{\alpha_{t}^{(t)} + {x_{t}^{T}{\hat{\beta}}_{t}^{(t)}}} = {\alpha_{t}^{(t)} + {\sum\limits_{i = 1}^{n}{{\hat{\beta}}_{t}^{({t,i})}x_{t}^{(i)}}}}}$with the parameters α_(t) ^((t)) and {circumflex over (β)}_(t) ^((t,i))computed on Step 1.

-   -   Step 3. Compute the cross validation statistic Q_(CV) as the        estimate of the distance between the return vector y and        predicted return vector ŷ^((t)) in a certain norm Q_(CV)=∥ê₁        ⁽¹⁾, . . . , ê_(N) ^((N))∥, ê_(t) ^((t))=y_(t)−ŷ_(t) ^((t)).

For example, for the quadratic problem (15), the cross validationstatistic Q_(CV), can be defined as follows:

$\begin{matrix}{{{{J^{(t)}\left( {{\overset{\_}{\beta}}_{1},\ldots\mspace{11mu},{\overset{\_}{\beta}}_{N}} \right)} = {{\sum\limits_{{s = 1},{s \neq t}}^{N}\left( {y_{s} - \alpha_{s} - {x_{s}^{T}\beta_{s}}} \right)^{2}} + {\lambda_{0}{\sum\limits_{s = 2}^{N}\left( {\alpha_{s} - \alpha_{s - 1}} \right)^{2}}} + {\lambda_{1}{\sum\limits_{s = 2}^{N}{\left( {\beta_{s} - {V_{s}\beta_{s - 1}}} \right)^{T}U_{s}\left( {\beta_{s} - {V_{s}\beta_{s - 1}}} \right)}}}}},\mspace{79mu}{\left( {{\overset{\hat{\_}}{\beta}}_{1}^{(t)},\ldots\mspace{11mu},{\overset{\hat{\_}}{\beta}}_{N}^{(t)}} \right) = {\underset{({\beta_{1},\ldots,\beta_{N}})}{\arg\;\min}{J^{(t)}\left( {{\overset{\_}{\beta}}_{1},\ldots\mspace{11mu},{\overset{\_}{\beta}}_{N}} \right)}}}}\mspace{14mu}\mspace{79mu}{{{subject}\mspace{14mu}{to}\mspace{14mu}{general}\mspace{14mu}{constraints}\mspace{14mu}(12)};}\mspace{79mu}{{{\hat{y}}_{t}^{(t)} = {\alpha_{t}^{(t)} + {x_{t}^{T}{\hat{\beta}}_{t}^{(t)}\mspace{14mu}({prediction})}}},\mspace{79mu}{e_{t}^{(t)} = {y_{t} - {{\hat{y}}_{t}^{(t)}\mspace{14mu}\left( {{prediction}\mspace{14mu}{error}} \right)}}},\mspace{79mu}{Q_{CV} = {{e_{\lbrack{1,N}\rbrack}^{(t)}}.}}}} & (25)\end{matrix}$

For example, the quadratic norm can be used

$\begin{matrix}{{Q_{{CV},{sq}} = {{{{\hat{e}}_{1}^{(1)},\ldots\mspace{11mu},{\hat{e}}_{N}^{(N)}}}_{sq}^{2} = {\frac{1}{N}{\sum\limits_{t = 1}^{N}\left( {y_{t} - {\hat{y}}_{t}^{(t)}} \right)^{2}}}}},} & (26)\end{matrix}$or the sum of absolute values of components

$\begin{matrix}{Q_{{CV},{abs}} = {{{{\hat{e}}_{1}^{(1)},\ldots\mspace{11mu},{\hat{e}}_{N}^{(N)}}}_{abs} = {\frac{1}{N}{\sum\limits_{t = 1}^{N}{{{y_{t} - {\hat{y}}_{t}^{(t)}}}.}}}}} & (27)\end{matrix}$

The statistics (26) and (27) can be further scaled to make themcomparable across different analyzed portfolios or instruments.

In another embodiment, the cross validation statistic can be measured asthe Predicted R-Squared Statistic PR²

$\begin{matrix}{{{PR}^{2} = {1 - \frac{Q_{{CV},{sq}}}{{Var}_{sq\eth}(y)}}},{{{Var}_{sq}(y)} = {\frac{1}{N}{\sum\limits_{t = 1}^{N}\left( {y_{t} - \overset{\_}{y}} \right)^{2}}}},} & (28)\end{matrix}$and the Predicted R-Statistic PR

$\begin{matrix}{{PR} = {1 - {\frac{{{{\hat{e}}_{1}^{(1)},\ldots\mspace{11mu},{\hat{e}}_{N}^{(N)}}}_{abs}}{{{{y_{1} - \overset{\_}{y}},\ldots\mspace{11mu},{y_{N} - \overset{\_}{y}}}}_{abs}}.}}} & (29)\end{matrix}$

Note that (28) is computed similar to the regression R-squaredstatistic.

3.2 Using CV in Parameter Selection

Note that the cross validation statistic Q_(CV) discussed above is afunction of the parameter vector Q_(CV)=Q_(CV)(λ), λ=(λ₀,λ₁, . . . ,λ_(m)). Choosing a different parameter vector λ for the objectivefunction (15) results, in general, in different solutions, differentpredictions and, therefore, different prediction errors.

In one embodiment, the cross validation statistic described above can beused to determine optimal model parameters, by solving the followingoptimization problem:

$\lambda_{opt} = {\underset{\lambda}{\arg\;\min}{{Q_{CV}(\lambda)}.}}$

Such an optimal value of λ would produce the minimal prediction error.Note that the selection of λ through minimizing the prediction errormakes it a version of the James-Stein estimator producing the smallestprediction error, for example, as described in Stone, M.,Cross-validatory choice and assessment of statistical predictions.Journal of Royal Statistical Soc., B 36, 1974, pp. 111-133.

3.3 Using CV in Model Selection.

The following three methods can be used, either independently or incombination with each other, in order to obtain the optimal set offactors in the optimization problem (16,20) by using Q_(CV) statisticdefined in 3.1.

-   -   a) A solution of the optimization problem (16) and the Q_(CV)        statistic are computed for each subset Ω of n′ factors. The        optimal set of factors is chosen as one that results in the        minimum value of Q_(CV) (Ω) across all such subsets:

$\Omega_{opt} = {\underset{\Omega}{\arg\;\min}{{Q_{CV}(\Omega)}.}}$

-   -   b) Forward stepwise selection by adding factors one-by-one until        no further improvement in the cross-validation measure is        possible or a certain threshold Q_(CV) statistic increment is        met.    -   c) Backward stepwise selection by starting with all factors and        removing factors one-by-one until further improvement in the        cross-validation statistic is not possible or the threshold        Q_(CV) statistic increment value is met.

4. Additional Embodiments

The methodology set forth above provides an efficient manner for one ormore processors housed in computer systems, such as in serverscommunicating with end-user terminals via an internet or intranetconnection) to provide the end-user with information that mostaccurately estimates the time-varying factor exposures (or otherweights) for factors (or other independent variables) in their model, bysolving the constrained multi-factor dynamic optimization problem,determining structural breakpoints for the factor exposures or weights,and assisting the end-user determine how to model the behavior of thedependent financial or economic variable with the use of the crossvalidation statistic. The processors can be programmed with algorithmsfollowing the steps set forth above, retrieving data related to theindependent variables and dependent financial or economic variables fromdatasets stored in databases that are housed in either the same computersystem, or in a different computer system, communicating with the one ormore processors in order to obtain the requested estimates.

The methodology can be programmed into a computer by being incorporatedinto computer readable program code embodied in a computer usable medium(e.g., a disk), in order to create a computer program product that isused to evaluate the model for the dependent financial or economicvariable. The information from the computer program product can be usedto evaluate the performance of an asset collection based on theinformation generated from the model.

The methodology can be implemented in an article of manufacture thatincludes an information storage medium encoded with a computer-readabledata structure adapted for use in evaluating the performance of an assetcollection over the Internet or other connection. The data structureincludes data fields with information relating to each aspect of themodel or problem, such as, for example: information related to thereturn on the asset collection (or other dependent financial or economicvariables) over the period of time; information related to the factors(or other independent variables) over the period of time; informationrelated to the factor exposures (or weights for the independentvariables) over the period of time, determined through the solution ofthe constrained multi-factor dynamic optimization problem; informationrelated to the structural breakpoint ratios for the factor exposures orweights; and information related to the cross validation statistic ofthe model.

The article of manufacture can include a propagated signal adapted foruse in a method of estimating time-varying factor exposures (or otherweights) in the dynamic optimization model through the period of time.The method includes one or more of the principles, formulations andsteps set forth above, and the signal is encoded with informationrelating to the various aspects of the model or problem.

The information generated by such dynamic optimization models orproblems can be used, for example, to evaluate the performance of amutual fund, the management of a portfolio or the sensitivity of acertain security or instrument or class of securities or instruments tovarious economic or financial indexes or indicators.

5. Conclusion

In the preceding specification, the present invention has been describedwith reference to specific exemplary embodiments thereof. Although manysteps have been conveniently illustrated as described in a sequentialmanner, it will be appreciated that steps may be reordered or performedin parallel. It will further be evident that various modifications andchanges may be made therewith without departing from the broader spiritand scope of the present invention as set forth in the claims thatfollow. The description is accordingly to be regarded in an illustrativerather than a restrictive sense.

1. A method for determining at least one factor exposure of an assetcollection for each of a plurality of time intervals in a period oftime, the asset collection including at least one asset, comprising:receiving performance data of the asset collection for the plurality oftime intervals; receiving data related to at least one factor for theplurality of time intervals, the at least one factor influencing theperformance data; for each of the plurality of time intervals,determining an objective function which includes an estimation errorterm and at least one transition error term, the estimation error termrepresenting an estimation error at each time interval between theperformance of the asset collection and a sum of products of each of theat least one factor exposure and its respective factor, the at least onetransition error term representing a transition error at each timeinterval after a first time interval for each of the at least one factorexposure between the time interval and a prior time interval; and foreach of the plurality of time intervals, determining the at least onefactor exposure by optimizing a value of the objective function, whereineach of the step of determining the objective function and the step ofdetermining the least one factor exposure is performed using at leastone processor.
 2. The method according to claim 1, wherein theperformance data including at least one of (a) a price of the assetcollection, (b) a return of the asset collection and (c) a function ofat least one of the price and the return.
 3. The method according toclaim 1, further comprising: defining at least one constraint on the atleast one factor exposure for at least one of the plurality of timeintervals.
 4. The method according to claim 3, wherein the at least oneconstraint is a bound constraint.
 5. The method according to claim 3,wherein the at least one constraint is a general linear constraintrelating to a structure of the factor exposures in at least one of theplurality of time intervals.
 6. The method according to claim 5, whereinthe general linear constraint is a budget constraint, the budgetconstraint requiring estimated factor exposures for a subset of factorsin the model to sum to one, the subset including at least one of thefactors.
 7. The method according to claim 6, wherein the defining stepincludes the substeps of: establishing the budget constraint for the atleast one exposure factor for a subset of the factors in the model,establishing the factors excluded from the subset as the factorsrepresenting the hedging instruments in the asset collection, andestablishing a hedging constraint on the possible values of the factorexposures for a the factors in the subset of hedging instruments inrelation to the factor exposures for at least one factor included in thein the budget constraint of the asset collection.
 8. The methodaccording to claim 5, wherein the general linear constraint is a budgetconstraint, the budget constraint requiring estimated factor exposuresfor all the factors to sum to one.
 9. The method according to claim 1,wherein the determining the objective function step includes the substepof: formulating the objective function as a parameter-weighted sum, theparameter-weighted sum being a sum of a norm of the estimation errorterm and a parameter-weighted norm of each of the at least onetransition error term.
 10. The method according to claim 1, wherein thedetermining the objective function step includes the substep of:formulating the objective function as a parameter-weighted sum, theparameter-weighted sum being a sum of a quadratic norm of the estimationerror term and a parameter-weighted quadratic norm of each of the atleast one transition error term.
 11. The method according to claim 10,wherein the determining step includes the substeps of: reformulating theobjective function without the at least one constraint, the reformulatedobjective function without the at least one constraint being equivalentto the objective function with the at least one constraint anddetermining the at least one factor exposure by minimizing the value ofthe reformulated objective function using one of a Flexible LeastSquares function and a Generalized Flexible Least Squares function. 12.The method according to claim 1, further comprising: determining a crossvalidation statistic of the model by testing an accuracy of a predictedperformance of the asset collection for at least one of the plurality oftime intervals.
 13. The method according to claim 12, wherein thetesting step includes the step of: calculating a prediction error of theperformance of the asset collection at each tested time interval as adifference between an actual performance of the asset collection and apredicted performance of the asset collection at the tested timeinterval, wherein the predicted performance of the asset collection ateach tested time interval is determined as a function of predictedfactor exposures and their respective factors at the tested timeinterval.
 14. The method according to claim 13, wherein the step ofdetermining the predicted performance includes the substeps of: (A)creating, for each tested time interval, a reduced dataset to determinethe predicted performance of the asset collection at the tested timeinterval, the reduced dataset excluding information relating to theactual performance of the asset collection at the tested time interval,and (B) determining, with the reduced dataset for the tested timeinterval, the predicted factor exposures minimizing the value of theobjective function.
 15. The method according to claim 12, furthercomprising: determining a value of a parameter, weighting the norm of atleast one transition error term in the parameter-weighted sum, as afunction of the cross validation statistic of the model, wherein theparameter-weighted sum is one of (a) a sum of a norm of the estimationerror term and a parameter-weighted norm of each of the at least onetransition error term and (b) a sum of a quadratic norm of theestimation error term and a parameter-weighted quadratic norm of each ofthe at least one transition error term.
 16. The method according toclaim 12, further comprising: selecting the at least one factor of themodel as a function of the cross validation statistic of the model. 17.The method according to claim 1, wherein the step of determining theobjective function includes the substep of: formulating the at least onetransition error term as a function determining, at each time interval,a difference between a value of an estimated factor exposure for itsrespective factor at the time interval, and a value of the factorexposure induced by a change in the respective factor between the timeinterval and a prior time interval.
 18. The method according to claim 1,further comprising: selecting one of a security and a portfolio ofsecurities as the asset collection for the model.
 19. A method fordetermining at least one factor exposure of an asset collection for eachof a plurality of time interval in a period of time, comprising: (A)receiving performance data of the asset collection for the plurality oftime intervals, the asset collection including at least one asset; (B)receiving data related to at least one factor for the plurality of timeintervals, the at least one factor influencing the return; (C) for eachof the plurality of time intervals, determining an objective function,the objective function including an estimation error term and at leastone transition error term, the estimation error term representing anestimation error at each time interval between the performance of theasset collection and a sum of products of each of the at least onefactor exposure and its respective factor, the at least one transitionerror term representing a transition error at each time interval after afirst time interval for each of the at least one factor exposure betweenthe time interval and a prior time interval, wherein the step (C)includes the substep of formulating the objective function as aparameter-weighted sum, the parameter-weighted sum being a sum of aquadratic norm of the estimation error term and a parameter-weightedquadratic norm of each of the at least one transition error term; (D)for each of the plurality of time intervals, determining the at leastone factor exposure by optimizing a value of the objective function; and(E) determining a cross validation statistic of the model by testing anaccuracy of a predicted performance of the asset collection at each timeinterval, the testing step includes the substep of calculating aprediction error of the performance of the asset collection at eachtested time interval as a difference between an actual performance ofthe asset collection and a predicted performance of the asset collectionat the tested time interval, wherein the (E) step includes the substepof calculating a quadratic norm of all the prediction errors of theperformance of the asset collection over the period of time, the substepof calculating the quadratic norm including the step of summing up overall tested time intervals the square of a difference between the actualperformance of the asset collection and the predicted performance of theasset collection, and dividing the sum by a number of the tested timeintervals in the period of time, and wherein each of the C, D and Esteps is performed using at least one processor.
 20. The methodaccording to claim 19, further comprising the step of: defining at leastone constraint of the at least one factor exposure for at least oneinterval.
 21. A computer readable storage medium including a set ofinstruction executable by a processor for determining at least onefactor exposure of an asset collection for each of a plurality of timeintervals in a period of time, the asset collection including at leastone asset, the set of instructions operable to: receiving performancedata of the asset collection for the plurality of time intervals;receiving data related to at least one factor for the plurality of timeintervals, the at least one factor influencing the performance data; foreach of the plurality of time intervals, determining an objectivefunction which includes an estimation error term and at least onetransition error term, the estimation error term representing anestimation error at each time interval between the performance of theasset collection and a sum of products of each of the at least onefactor exposure and its respective factor, the at least one transitionerror term representing a transition error at each time interval after afirst time interval for each of the at least one factor exposure betweenthe time interval and a prior time interval; and for each of theplurality of time intervals, determining the at least one factorexposure by optimizing a value of the objective function.